Without a calculator, find $\cos\frac{\pi}{3}$

Question

Without a calculator, find $\cos\frac{\pi}{3}$.

I started AP Calc today. I’m a sophomore and I skipped precalc and, well, I am utterly confused.

Can someone please teach me how to solve this from the basics by hand? Thank you so much!

Answer 1

HINT

Consider an equilateral triangle $\triangle ABC$ and draw its height $\overline{AH}$ related to the side $\overline{BC}$, which is also the median. Thus the cosine is given by \begin{align*} \cos\left(\frac{\pi}{3}\right) = \frac{HC}{AC} = \frac{1}{2} \end{align*}

Answer 2

The "end result" is 1/2, and this is my "quick route"

We have π radian = 180 degree, so π/3 radian = 60 degree. 60 (degree) + 30 (degree) = 90 (degree) --> A right angle. We also have 30 degree = π/6 (at this point, you should know that "rad" can be conveniently omitted)

Therefore, $cos(π/3)=sin(π/6)$. The side opposite to the 30 degree angle in a triangle is half as long as the hypotenuse, or $sin(π/6)= 1/2$.

Then, we have $cos(π/3)=1/2$

Answer 3

On the one hand, the double-angle identity implies $$\cos(\pi/3)=\cos(2(\pi/6))=1-2\sin(\pi/6)^2.$$ But we also know (sine/cosine of complementary angles) that $$\cos(\pi/3)=\sin(\pi/2-\pi/3)=\sin(\pi/6).$$

Hence $\cos(\pi/3)=1-2\cos(\pi/3)^2.$ One solution to this is $\cos(\pi/3)=-1$, which is obviously not correct. That leaves the second solution, which is $\cos(\pi/3)=1/2.$